93 problems

Ito's conjecture. The group $Q_{8t}$ contains such a relative difference set for all positive integers $t$.

Stanchescu's conjecture. The extremal lower bound for every finite, non-empty $d$-dimensional set $A\subseteq\mathbb{R}^d$ is

Let $q$ be a number, and let $G$ be an Abelian group with … A subset $B\subset G$ is a difference set if every non-unit element of $G$ has a unique representation $g=ab^{-1}$ with…

Let a finite Sidon set be a finite set of integers in which every difference between two distinct elements has at most one representation as an ordered difference of elements of th…

Let $m$ be an odd positive integer and let $f$ be one of the eleven trinomials specified in the paper. For the punctured value-set $D(f)^*$ associated with $f$ over the binary fiel…

Conway's conjecture. Every set $A\subset \mathbb Z$ satisfies

Let $m\geq 5$ be odd, let $D_f$ be defined from one of the eleven trinomials in Ding's difference-set conjecture, and let ${\mathcal C}_{D_f}$ be the associated binary linear code.…

Zero-dimensionality conjecture. $L_m$ is zero-dimensional for every even $m\geqslant24$.

Let $m$ be an odd integer with $m>7$ and $m\not\equiv 0 \pmod{3}$. In the additive group $({\mathbb F}_{3^m},+)$, let ${\rm P}$, ${\rm DY}(1)$, ${\rm DY}(-1)$, ${\rm RT}(1)$,…

Let $G$ be a finite abelian group, and let $D\subseteq G$ be a skew Hadamard difference set, meaning that $G$ is the disjoint union of $D$, $D^{(-1)}=\{d^{-1}\mid d\in D\}$, and th…

Let $j$ and $k$ be nonnegative integers with $k$ even. Prescribed-defect existence conjecture. There exists a positive integer $n$ and a set $S\subset\{0,1,\dots,n-1\}$ such that…

For $n\ge 1$, let $S$ range over subsets of $\{0,1,\dots,n-1\}$. For nonnegative integers $j$ and $k$, define … assuming the limit exists. Since $S-S$ is symmetric about $0$, only…

Let $m>9$ for the HKM difference sets and $m>7$ for the Lin difference sets, and let $\text{epsilon}(m)$ and $\text{delta}(m)$ be the quantities appearing in the formulas for the n…

Cyclic Hadamard difference-set conjecture. No cyclic Hadamard difference sets exist other than those in these three families.

Ryser's conjecture. No cyclic difference set exists with

Let $p$ be a prime, and let $\mathcal{R}_p$ denote the set of nonzero squares in ${\mathbb F}_p$. Lev and Sonn's conjecture. For all sufficiently large primes $p$, there is no subs…

Let $p$ be prime, and let $K$ be the multiplicative subgroup of $\ell$-th powers in $\mathbb{F}_p$. A power difference set conjecture asserts that if $K$ forms a difference set of…

Let $D$ be a $(v,k,\lambda)$ difference set, let $t$ be an integer relatively prime to $v$, and let $n_1$ be a divisor of $n$ relatively prime to $v$. Suppose that for each prime d…

Erdős's conjecture. For every positive integer $\ell$, with finitely many exceptions, there exists a non-trivial perfect system of difference sets whose positive differences are ex…

Bermond's conjecture. A $(v,4,1)$-PDF exists for every $v\equiv 1\pmod{12}$ with $v\geqslant 49$.

A cyclic projective plane is a projective plane admitting a regular action by a cyclic group. A projective plane is Desarguesian when it satisfies Desargues's theorem, and a perfec…

Reversibility conjecture for difference sets. Difference sets are conjectured to be reversible only in very limited circumstances.

McFarland's conjecture. Every reversible abelian difference set has parameters of one of these forms.

Let a difference set have parameters matching those of the complement of a classical Singer difference set, and suppose the Singer difference set has even dimension. A lifting is a…

Let $p$ be an odd prime and let $g$ be a natural number. Define … Theorem 1 establishes that these limits exist and are positive real numbers. Even–odd limit conjecture. The two li…